ICS220 – Data Structures and Algorithms Dr. Ken Cosh Week 5.
ICS 220 – Data Structures and Algorithm Analysis Dr. Ken Cosh Week 4.
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Transcript of ICS 220 – Data Structures and Algorithm Analysis Dr. Ken Cosh Week 4.
Last Week
• Linked Lists– Singly, Doubly, Circular.– Skip Lists– Self Organising Lists– Double Ended Queues (or Deque)
Stacks
• Stacks are linear data structures, that can only be accessed at one end for storing and retrieving data.
• New data is added to the top of a stack, and data is also retrieved from the top of the stack.
• Similar to a stack of trays in a canteen.
• It is a LIFO structure (Last In First Out).
Queues
• Queues are also linear data structures, however it is a waiting line, where both ends are used.
• Data is added to one end of the line, and retrieved from the other.
• Similar to a Queue in a bank etc.
• It is a FIFO structure (First In First Out).
Stacks
• Key operations;– Clear() – clears the stack– isEmpty() – Tests if the stack is empty– Push(el) – adds ‘el’ to the top of the stack– Pop() – Retrieves the top element of the stack– topEl() – Returns the top element without
removing it.
Stack Use
• Consider the problem of matching delimiters in a program;– Delimiters : [, ], {, }, (, ), /*, */
• Problem; to test the delimiters have been correctly matched;– A) while(m<(n[8] + o)) {p=7; /*initialise p*/ r=6;}– B) a = b + ( c – d ) * ( e - f ))
• Case A should return a success, while case B should return an error.
Stack Case
• A) while(m<(n[8] + o)) {p=7; /*initialise p*/ r=6;}
– Add to stack ( - ( – Add to stack ( - ( (– Add to stack [ - ( ( [– Remove from stack [ - ( (– Remove from stack ( - (– Remove from stack ( -– Add to stack { - {– Add to stack /* - { /*– Remove from stack */ - }– Remove from stack } -
Implementing as a Vector#ifndef STACK#define STACK#include <vector>template<class T, int capacity = 30>class Stack{public:
Stack() { pool.reserve(capacity); }void clear() { pool.clear(); }bool isEmpty() const { return pool.empty(); }T& topEl() { return pool.back(); }T pop() {
T el = pool.back();pool.pop_back();return el; }
void push(const T& el) { pool.push_back(el); }private:
vector<T> pool;};
#endif //STACK
Implementing as a Linked List#ifndef LL_STACK#define LL_STACK#include <list>template<class T>class LLStack {public:
LLStack() { }void clear() { lst.clear(); }bool isEmpty() const { return lst.empty(); }T& topEl() {return lst.back(); }T pop() {
T el = lst.back();lst.pop_back();return el; }
void push(const T& el) { lst.push_back(el); }private:
list<T> lst;};#endif // LL_STACK
Comparison
• The linked list matches the stack more closely – there are no redundant ‘capacity’.
• In the vector implementation the capacity can be larger than the size.
• Neither implementation forces the program to commit to the size of the stack, although it can be predicted in the vector implementation.
• Pushing and Popping for both implementations is in constant time; O(1).
• Pushing in the vector implementation when the capacity is full requires allocating new memory and copying the stack to the new vector; O(n).
Queue
• Key Operations;– Clear() – Clear the queue– isEmpty() – Check if the queue is empty– Enqueue(el) – Add ‘el’ to end of queue– Dequeue() – Take first element from queue– firstEl() – Return first element without
removing it.
Queue Use
• Simulating any queue;– To determine how many staff are needed in a
bank to maintain a good level of service,– Or, how many kiosks to open at the motorway
toll.
Option 1 - Array
• The obvious problem with using an array is that as you remove elements from the front of the queue, space then becomes wasted at the front of the array.
• This can be avoided using a ‘circular array’, which reuses the first part of the array.
5 6 7?
5
67
Circular Array• As elements at the front of the array are
removed those cells become available when the array reaches the end.
• In reality a circular array is simply a one dimensional array, where the enqueue() and dequeue() functions have the extra overhead of;– Checking if they are adding / removing the element in
the last cell of the array.– Checking they aren’t overwriting the first element.
• Therefore the circular array is purely a way of visualising the approach.
• The code on the next slides demonstrates some of the functions you might need if you chose to implement using an array.
Queue – Circular Array#ifndef ARRAY_QUEUE#define ARRAY_QUEUEtemplate<class T, int size = 100>class ArrayQueue {public:
ArrayQueue() { first = last = -1; }void enqueue(T);T dequeue();bool isFull() { return first == 0 && last == size-1 || first == last +1; }bool isEmpty() { return first == -1 }
private:int first, last;T storage[size];
};
Queue – Circular Array cont.template<class T, int size>void ArrayQueue<T,size>::enqueue(T el) {
if (!isFull())if (last == size-1 || last == -1) {
storage[0] = el;last = 0;if (first == -1)
first = 0;}else storage[++last] = el;
else cout << “Full queue.\n”;}template<class T, int size>T ArrayQueue<T,size>::dequeue() {
T tmp;tmp = storage[first];if (first == last)
last = first = -1’else if (first == size -1)
first = 0;else first++;return tmp;
}#endif //ARRAY_QUEUE
Option 2 – Doubly Linked List
• A perhaps better implementation uses a doubly linked list.– Both enqueuing and dequeuing can be
performed in constant time O(1).– If a singly linked list was chosen then O(n)
operations are needed to find the ‘other’ end of the list either for enqueuing or dequeuing.
Doubly Linked List#ifndef DLL_QUEUE#define DLL_QUEUE#include <list>template<class T>class Queue {public:
Queue() { }void clear() { lst.clear(); }bool isEmpty() const { return lst.empty(); }T& front() { return lst.front(); }T dequeue() { T el = lst.front();
lst.pop_front();return el; }
void enqueue(const T& el) { lst.push_back(el); }private:
list<T> lst;};#endif // DLL_QUEUE
Priority Queues
• Queuing is rarely that simple!• What happens when a police car approaches a
toll point? Or a disabled person visits a bank? Or in fact many of the queuing situations in Thailand?
• A standard queue model won’t effectively model the queuing experience.
• In priority queues elements are dequeued according to their priority and their current queue position.
Queue Theory
• There has been much research into how to best solve the priority queuing problem – if you are interested simply look up “Queue Theory”.
Linked List Variation
• We can use a linked list to model the new queue, by simply making a simple variation. There are 2 options; – When adding a new element to the list, search
through the list to place it in the appropriate position – O(n) for enqueue().
– When removing an element, search through the list to find the highest priority element – O(n) for dequeue().
Alternative
• Have a short ‘ordered’ list, and a longer unordered list.
• Priority elements are added to the ordered list, non-priority elements are in the longer list.
• From this model, dequeue() is of course constant O(1), but enqueue() can be O(√n) with maximised efficiency.
(Blackstone 1981)
STL - Stack
• Stack exists in the STL, with the following key member functions;
bool empty() const – returns true if stack is empty.
void pop() – removes the top element
void push(const T& el) – insets el to the top of the stack
size_type size() const – returns the size of the stack
stack() – constructor for empty stack
T& top() – returns top element from stack
STL - Queue
• Queue exists in the STL, with the following key member functions;
T& back() – returns last element
bool empty() const – returns true if queue empty
T& front() – returns first element in queue
void pop() – remove first element in queue
void push(const T& el) – insert el at back of queue
queue() – constructor for empty queue
size_type size() const – returns size of queue
Programming Assignment
• A Queue or Stack can be used to perform calculations on very large integers. There is a limit to the size of integers, so performing the following calculation can be difficult;– 1344823508974534523+23472347094730475
• Write a program that can perform the 4 key arithmetic operations, +, -, *, /, on very large integers, returning an integer.